Splitting lemma

In mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements for a short exact sequence are equivalent.

Given a short exact sequence with maps or arrows, q and r, between the category objects:

one writes the additional arrows t and u for maps that may not exist:

Then the following statements are equivalent:

If these statements hold, the sequence is called a split exact sequence, and the sequence is said to split.

In the above short exact sequence, where the sequence splits, it allows one to refine the first isomorphism theorem, which states that:

to:

where the first isomorphism theorem is then just the projection onto C.

It is a categorical generalization of the rank–nullity theorem (in the form V ≅ ker T ⊕ im T) in linear algebra.

First, to show that 3. implies both 1. and 2., we assume .3 and take as t the natural projection of the direct sum onto A, and take as u the natural injection of C into the direct sum.

To prove that 1. implies 3., first note that any member of B is in the set (ker t + im q). This follows since for all b in B, b = (b − qt(b)) + qt(b); qt(b) is obviously in im q, and b − qt(b) is in ker t, since

Next, the intersection of im q and ker t is 0, since if there exists a in A such that q(a) = b, and t(b) = 0, then 0 = tq(a) = a; and therefore, b = 0.

...
Wikipedia